I start my car, everything sounds fine.
I start to drive and here a clucking sound.
Sounds like some thingis broke, or stuck and clunking somewhere underneath the car.
While moving about 20 mph I put the clutch in turn off the engine and roll to a stop.
I continue to hear the grinding until the car stop rolling.

Any ideas on what the noise might be?

 

What year Civic?

Is the clunking sound increased when turning right or left?

You might want to inspect your front axles.

 

checkyour splash guards ,mine came loose on one side and was rubbing the tire it made terrible noises at low speeds

 

did you put new rims on it?
ive seen people get new rims and not have the lugnuts tight and the wheele wobble

 

Brakes rubbing (kinda like a grinding metal sound)! or rotors warped? I haven't driven mine in a while, so the brakes need replaced and they rub!

 

i'd say you have a small chicken in ur engine bay or under ur dash [sm=happybounce.gif]